It's not the thing that changes, just how we describe it.

These titanium (Ti) bars each have a fixed length. To within a few layers of atoms, we could measure that length, and we'd expect it never to change.*

Two methods

There are two most-used methods of unit conversion, but first, a word on why we use those at all. After all, if you want to change 27 inches into millimeters, you can always go to Google or some other helpful website or app and just ask it to do the calculation for you. Here's a screen shot of a Google™ conversion:

1. Big parentheses

I begin with this method because it's the one I use most often. We'll learn it by example, and we'll begin with a very simple one.

We know that there are 100 centimeters (cm) in every one meter (m), so let's ask the question: How many meters is 123 cm? Now this is a pretty easy question to do just by looking at it, knowing your metric prefixes and moving a decimal place, but let's do the unit conversion anyway.

We begin by writing the number we know with its units: 123 cm.

$$123 \; cm$$

Next to it, on the right, write a big open parenthesis with an empty fraction inside.

$$123 \, cm \left( \frac{\phantom{0000}}{\phantom{0000}} \right)$$

Now we know that our final answer should have units of meters (m), so we want to "get rid" of the units cm. By "get rid of" we mean to treat the unit cm as an algebraic variable, and divide it by itself to get 1, a trivial multiplier that changes nothing. So our units (neglecting numbers for now) will look like this:

2. Grid method or "factor labeling"

The grid method, often called factor labeling, is sometimes preferred to the parentheses method above, though it is really just the same, except for some added structure.

Here's an example. Let's convert 1322 milliliters (ml) to liters (L), noting that there are 1000 ml in every 1 L.

We begin as above (and as always), by writing down the number we have with its units:

Next, construct a grid of lines like this:

Next (we are really just following the steps in the box above), write down the units ml and L so that the unit we want to get rid of (ml) is across the horizontal bar (really a fraction bar) from the starting number:

Practice problems — one-step conversions

Caclulate the following conversions.

A length conversion tool

Here is a little length-unit conversion tool. You can use it to play around a bit with length units. For example, type 29028 (the height of Mt. Everest) into the top box, choose units of feet, then choose units of meters in the bottom box. You'll get the height of Everest in meters. I've also included units of fathoms (used to measure deep ocean depths) and furlongs (lengths of horse-racing tracks) just for fun. Enjoy.

More complicated conversions: chaining

Sometimes we have to do unit conversions that require more than one step. Let's look at one conversion and perform it four different ways.

Convert 25 meters per second (m/s) to miles per hour (mi./h).

First we'll do this conversion by performing two separate unit conversions, meters to miles (both units of length), and then seconds to hours (both units of time).

Method 1 — parentheses

We begin by writing our number with its units. Write m/s as meters over seconds, not with a diagonal slash. Those can be confusing later.

$$25 \, \frac{m}{s}$$

Add an empty fraction in parentheses:

$$25 \, \frac{m}{s} \left( \frac{\phantom{0000}}{\phantom{0000}} \right)$$

Write in the units. We have a choice here. We can only change one unit at a time, so let's chose miles. that means that meters will go in the denominator so that it divides (to 1) the meters in the numerator:

$$25 \, \frac{m}{s} \left( \frac{\phantom{00} mi.}{\phantom{000} m} \right)$$

Now recognize that $1 \, mi. = 1609 \, m.$ That's something you could look up if you don't know it.

$$\require{ cancel } 25 \, \frac{\cancel{m}}{\color{#E90F89}{s}} \left( \frac{1 \, \color{#E90F89}{mi.}}{1609 \cancel{m}} \right)$$

Canceling units gives us new units of miles/s, and following the arithmetic (remember, that 25 is really 25/1), we get the speed in miles per second:

$$= \frac{25}{1609} = 0.01554 \, \frac{mi.}{s}$$

Now we have to convert seconds to hours with a new beginning point, 0.01554 mi./s (I'm keeping a lot of digits here in the middle of the calculation to avoid round-off errors. In the empty fraction we'll put seconds in the numerator.

$$0.01554 \, \frac{mi.}{s} \left( \frac{\phantom{000}s}{\phantom{000}h} \right)$$

There are 3600 seconds in 1 hour, so our conversion is:

$$0.01554 \, \frac{mi.}{s} \left( \frac{3600 \, s}{1 \, h} \right)$$

Canceling units (seconds) gives us our goal units of mi./h (sometimes written as mph):

Example – squared units

An acre is a measure of land area used mainly in the U.S. It is a square with sides of length 208.71 ft. Given that there are 2.54 cm in every inch, calculate the area of an acre of land in square meters (m²)

Solution: Let's convert an area to an area, so we'll first calculate the number of square feet (ft²) in an acre:

$$(208.71 \, ft.)(208.71 \, ft.) = 43,650 \, ft^2$$

Now we're going to convert feet to inches (ignoring the fact that we're working with square feet for now). The way I remember that is that there are 12 inches in every foot and 2.54 cm in every inch. Don't ask me why I remember that; I just do. You can look up the number of feet in a meter if you want.

But we are working with square feet, and therefore we'll need square meters. Do arrive at that, we can take our entire conversion of feet to meters, and square it. Notice that everything will cancel nicely and we'll end up with units of m².

Example – cubed units

Convert the volume 12,000 gallons to cubic meters (m³)

Solution: To do this conversion, we need one important piece of information, the link between gallons and liters:

In the metric system, the liter is defined as one cubic decimeter, were a decimeter (remember that "deci" means "tenth") is a tenth of a meter. So we have all the links we need to do this unit conversion. Here it is:

$$\require{ cancel } 12,000 \cancel{gal} \left( \frac{3.7854 \cancel{L}}{1 \cancel{gal}} \right) \left( \frac{1 \cancel{dm^3}}{1 \cancel{L}} \right) \left( \frac{1 \, m}{10 \cancel{dm}} \right)^3$$

Notice that the term (1 m/10 dm) is cubed. This must be so to cancel the dm³ in the term before it, and we must cube the entire term in the parentheses. The result of cubing that term shows us that there are 1000 (10³) cubic decimeters (or liters) in every cubic meter. The diagram illustrates that.

The result of our calculation is that 12,000 gallons is 45.4 m³.

Video examples

Example 1

Two introductory examples of unit conversion, done two ways.

Minutes of your life: 4:42